TheKoiMaiden is doing a bang up job helping you out, but just to add in some points...
You can speed up/help the cycle by doing a couple additional things here. If you know anyone that has an establish, HEALTHY tank, you can borrow some "bioseed" to add to your tank. "bioseed" can mean anything on which beneficial bacteria can grow (filter media, substrate, decorations, plants). This addition can seed your tank with bacterial and cut down on the time it takes you to multiple your own bacteria. Just make sure you only use seed from a healthy tank, no diseases or illness (past or present)!
Another thing to help is to add plants. With goldies being the little plant eating machines they are, this can either work or just ending being a snack. So choose plants carefully! I would reccommend two plants- duckweed and hornwort. (You can click on the shaded names to read more.) Both are fast growing, so they suck up (at least some of) the ammonia before it gets a chance to hurt your fish. Duckweed is a surface plant, and goldies do love to eat it. Its nutritious! So just make sure you get a large enough portion that they fish don't eat it all up before it gets a chance to start to multiple. Hornwort is a floating plant, but it can also be anchored. It does shed its needles annoyingly under lower light, so its a bit messy and you really have to keep up on vacuuming up the shed leaves (you should be vacuuming at least weekly, anyway). But even considering the shedding, hornwort is awesome because it grows quickly, can survive low light, and isn't usually eaten by goldies (unless they're desperate, and even then it usually grows quickly enough to survive the damage).
During cycling, it helps to feed sparingly. The less you feed, the less waste is being produced, so the less ammonia (both from fish waste and food waste) is being released into the water. Lower, but still in the comfort range, temps can help too by slowing metabolism.
I think you said you have gravel, right? This should be vacuumed at least weekly. Do you have a gravel vac? If not, one can be made for common materials. There's a tutorial on youtube.com for a gravel vac made from a plastic water bottle.
A great alternative to the 10g temp aquarium would be using a large plastic storage bin as a temp tank. You can treat it just like an aquarium, using your filter and substrate, etc. This will allow you to have a much larger temp home for CHEAP or FREE while you're saving your money for their permanent home. Its not pretty, but its better for the fish.
Good luck! And thanks for trying to look out for fish!
Another good goldie plant is anacharis. In my experience, they love to snack on it and it gives them a little bit of variety. :) Obviously not so good when you are trying to cycle, though, as it just makes them poop more.
Actually, the ammonia sucking abilities outweigh the increased waste, IMO. But there are other plants that suck ammonia just as well but aren't as appetizing, like hornwort. The other reason that I didn't recommend Anacharis is because it requires more light than hornwort in order to grow well, and therefore doesn't suck up as much ammonia in lower light.
If it were me, I'd stick to duckweed and hornwort. Other plants can be added later, if the OP wants, and has the needed light and conditions.
Mina is right. Anarcharis shows the best growth under a higher light. Duckweed and hornwort are the two plants that have given me the most growth with my goldfish (sometimes as much as a foot a week with the hornwort!). Water sprite and anarcharis (as great as they are) need better lighting than what comes with the hood.
Good point Olympe. You can feed them greens like kale and spinach as well as squashes like yellow squash and zucchini. Peas are great, too. They will eat them frozen/thawed or fresh. These will help cut down on the ammonia load, too. Foods that are high in protein like well normal fish food will make them produce more ammonia than veggies. So when you don't want to eat your veggies feed them to your goldies!
Really? I've always found my anacharis grows just fine in low light, even ambient light. Then again, in the goldie tank it has always been eaten before it could grow much. :p Maybe it is hardier here (it is actually illegal in New South Wales because it is a noxious pest).
There are a few varieties of Anarcharis, Elodea, Egeria, whateveritscalled. I can never keep up with the current common and latin names. I took a quick search and found Egeria densa, Elodea nuttallii, Elodea callitrichoides, andElodea crispa [which is a misnomer] which all look similar. So maybe we're each familiar with different varieties? The only one I've seen sold here in my part of the US, whichever one it is, has never grown well for me in very low or low light. In fact, I've never tried it again because I got so frustrated with it!
Whichever variety you have, I'm glad you're having good luck with it!